On the arithmetic of rational hypersurfaces in toric varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918163797508096 |
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| author | Grassi, Gianluca |
| author_facet | Grassi, Gianluca |
| contents | In the toric variety $\mathcal{T}$, with Cox ring graded by $°(z_{2i})=(1,-1,0)$, $°(z_{2i+1})=(1,0,-1)$ and $°(w_\pm)=(0,1,0),(0,0,1)$, we study hypersurfaces $\widetilde{X}^{2n}\subset\mathcal T$ of multidegree $(2d+1,-d,-d)$ over a field $k$. These are the strict transforms of odd-degree hypersurfaces in $\mathbb{P}^{2n+1}$ with multiplicity $d$ along two skew conjugate $n$-planes. We prove that $\widetilde{X}^{2n}$ is $k$-rational and birational to $\mathbb{P}^{2n}$; and derive result on the distribution of its rational points over numbers and finite field. The case $d=1$ recovers the even-dimensional Fermat cubic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the arithmetic of rational hypersurfaces in toric varieties Grassi, Gianluca Algebraic Geometry Primary 12F20, 12E10, 14E08, 14M20, 12F10, Secondary 14G05 In the toric variety $\mathcal{T}$, with Cox ring graded by $°(z_{2i})=(1,-1,0)$, $°(z_{2i+1})=(1,0,-1)$ and $°(w_\pm)=(0,1,0),(0,0,1)$, we study hypersurfaces $\widetilde{X}^{2n}\subset\mathcal T$ of multidegree $(2d+1,-d,-d)$ over a field $k$. These are the strict transforms of odd-degree hypersurfaces in $\mathbb{P}^{2n+1}$ with multiplicity $d$ along two skew conjugate $n$-planes. We prove that $\widetilde{X}^{2n}$ is $k$-rational and birational to $\mathbb{P}^{2n}$; and derive result on the distribution of its rational points over numbers and finite field. The case $d=1$ recovers the even-dimensional Fermat cubic. |
| title | On the arithmetic of rational hypersurfaces in toric varieties |
| topic | Algebraic Geometry Primary 12F20, 12E10, 14E08, 14M20, 12F10, Secondary 14G05 |
| url | https://arxiv.org/abs/2510.16773 |